paul j. steinhardt- a quintessential introduction to dark energy

8/3/2019 Paul J. Steinhardt- A Quintessential Introduction to Dark Energy

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A Quintessential Introduction to Dark Energy

Paul J. Steinhardt

Department of Physics, Princeton University, Princeton, NJ 08540, USA

Abstract

Most of the energy in the universe consists of some form of dark energy that is gravitationally

self-repulsive and that is causing the expansion rate of the universe to accelerate. The possible can-

didates are a vacuum energy density (or, equivalently, a cosmological constant) and quintessence,

a time-evolving, spatially inhomogeneous component with negative pressure. In this review, we

focus on quintessence and ideas on how it might solve the cosmic coincidence problem, how it

might be distinguished observationally from a cosmological constant, and how it may affect the

overall cosmic history of the universe.

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Introduction

The discovery of dark energy is one of the most surprising and profound discoveries in

the history of science. Consider some of its implications:

Most of the energy in the universe is not matter. For its first 300 years, physics has

focused on the properties of matter and radiation, including dark matter. Now we know

that they represent less than 30% of the composition of the universe. The rest consists of

something we know virtually nothing about.

Most of the energy in the universe is not gravitationally attractive. We are probably the

last generation to have been taught that gravity always attracts, a notion which has been

presented as a basic fact of nature for hundreds of years. We are now aware that gravity

can repel, as well. Of course, the possibility of self-repulsive forms of energy was there inEinsteins general theory of relativity since its inception, but this point was never generally

appreciated until now. We must rewrite the textbooks to explain that the gravitationally

self-attracting matter with which we are familiar is the minority in the universe today and

for the indefinite future.

We live at a special time in the history of the universe. The Copernican revolution

taught us that there is nothing special about our location in the universe. If space is

uniform, then should not the same be true for time? Hubbles discovery that the universe

is expanding taught us that the universe is evolving, but the notion had been that the

evolution has been steady over the last 15 billion years with no remarkable changes. We

now know that time is anti-Copernican. We live at a special moment in cosmic history,

the transition between a decelerating, matter-dominated universe and an accelerating, dark

energy dominated universe. The progressive formation of ever-larger scale structure and

increasing complexity that characterized the matter-dominated universe has reached an end,

and now the universe is headed towards a period that is ever-emptier and structureless.

The future (and perhaps the past) is determined by dark energy. Clearly, the immediate

future of the universe will be governed by dark energy which, depending on its nature, will

determine the rate of dilution and cooling of the matter and energy. But, perhaps dark

energy plays a more profound role in the history of the universe, determining our distant

past as well as our long-term future. We will discuss the recent proposal of a cyclic universe

(Steinhardt & Turok 2002a) in which dark energy plays a key role is an important part of

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the engine driving the periodic evolution of the universe.

Given the profound implications, finding the identity of dark matter has emerged as

one of the most important scientific challenges of the 21st century. The first evidence for

dark energy emerged in the mid-90s (Ostriker & Steinhardt 1995, Krauss & Turner 1995).

First, improved observations confirmed that the total mass density is probably less than

half of the critical density (Bahcall et al. 1995, Carlberg et al. 1996, Bahcall et al. 1998).

At the same time, combined measurements of the cosmic microwave background (CMB)

temperature fluctuations and the distribution of galaxies on large scales began to suggest

that the universe is flat, consistent with the standard inflationary prediction. The only way

to have a low mass density and a flat universe, as expected from the inflationary theory,

is if an additional, nonluminous, dark energy component dominates the universe today.

The dark energy would have to resist gravitational collapse, or else it would already have

been detected as part of the clustered energy in the halos of galaxies. But, as long as most

of the energy of the universe resists gravitational collapse, it is impossible for structure to

form in the universe. The dilemma can only be resolved if the hypothetical dark energy was

negligible in the past and, then, only after galaxies and larger scale structure formed, became

the dominant energy in the universe. According to general relativity, the only type of energy

with this property has negative pressure. This argument (e.g., Ostriker & Steinhardt 1995)

rules out almost all of the usual suspects, such as cold dark matter, neutrinos, radiation,and kinetic energy, because they have zero or positive pressure. Furthermore, according

to Einsteins equations, negative pressure implies cosmic acceleration. So, this analysis

anticipated the supernovae results (Perlmutter et al. 1998, Riess et al.) which have provided

direct evidence for acceleration.

Hence, there are numerous lines of evidence establishing that dark energy exists and

that it comprises nearly 70% of the energy density of the universe today. But, all of these

observations do little to inform us about what the dark energy is.

I. WHAT IS THE DARK ENERGY?

The two logical possibilities for dark energy are the cosmological constant and

quintessence. The cosmological constant was first introduced by Einstein for the purpose of

constructing a static model of the universe. The repulsive cosmological constant delicately

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fine-tuned to balance the gravitational attraction of matter (Einstein 1917). Today, the

cosmological constant is recognized as vacuum energy, an energy assigned to empty space

itself that has negative pressure and induces cosmic acceleration. It has the same value

everywhere in space for all time, and it is chemically inert. And, unlike Einsteins original

concept, the cosmological constant, if it exists, has not been fine-tuned to balance the mat-

ter. Instead, the vacuum energy is overabundant, causing the expansion of the universe to

accelerate. The cosmological constant is completely defined by one number, its magnitude.

Quintessence is a dynamical, evolving, spatially inhomogeneous component with negative

pressure (Caldwell et al. 1998). The term derives from the medieval word for fifth element;

according to some metaphysicians at the time, the universe consisted of earth, air, fire and

water, plus an additional all-pervasive, component that accounted for the motion of the

Moon and planets.) In the current context, quintessence would be the fifth dynamical

component that has influenced the evolution of the universe, in addition to the previously

known baryons, leptons, photons, and dark matter.

Quintessence is characterized by its equation-of-state w p/, where p is the pressure and

is the energy density. Most models have 0 w > 1 whereas a cosmological constant has

w precisely equal to 1. The smaller is the value of w, the greater is its accelerating effect.

Unlike a cosmological constant, the quintessential pressure and energy density evolve in

time, and w may also. Furthermore, because the quintessence component evolves in time, itis, by general covariance, necessarily spatially inhomogeneous. In some models, quintessence

also has a time-varying speed of sound that can enhance the effect of fluctuations on the

cosmic microwave background and large scale structure.

It should be emphasized that the quintessence explanation for the dark energy does not

explain the longstanding problem of the cosmological constant. Prior to the discovery of

dark energy, it had been presumed that some symmetry or cancellation mechanism causes

the vacuum energy to vanish altogether or to shrink a level where it is negligible small. If

the dark energy proves to be quintessence, we would need to invoke the same cancellation

mechanisms.

A common model of quintessence is the energy density associated with a scalar field Q

slowly rolling down a potential (V(Q)). The pressure of the scalar field, p = 12Q2 V(Q)

is negative if the field rolls slowly enough that the kinetic energy density is less than the

potential energy density. The ratio of kinetic-to-potential energy is determined by equation-

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of-motion for the scalar field:

Q + 3HQ + V(Q) = 0. (1)

This determines the equation of state

w p

= 12Q2 V(Q)12

Q2 + V(Q)(2)

Depending on the detailed form of V(Q), the equation-of-state w can vary between 0 and

1. For most potentials, w evolves slowly with time. The field is assumed to couple

only gravitationally to matter. The Q-energy density decreases with time as 1/a3(1+wQ), so

negative pressure corresponds to a density which decreases more slowly than 1/a3.

The spatial inhomogeneities in Q evolve over time due to the gravitational interaction

between Q and clustering matter (Caldwell et al. 1998). The perturbations are importantbecause they can leave a distinguishable imprint on the CMB and large-scale structure. To

determine how the perturbations evolve, specifying w is insufficient. One must know the

response of the component to perturbations. This can be defined by specifying the sound

speed cs as a function of wavenumber k or, alternatively, by specifying the equations-of

motion (from which the perturbative equations can be derived). Note that it is possible, in

principle, to have two fluids with the same w but different cs, which would lead to distinct

observational predictions, as discussed later in this paper.

For a scalar field, the equation-of-motion for the perturbations Q in synchronous gauge

is:

Q + 3H Q + (c2sk2 + a2V(Q))Q =

1

2hkQ, (3)

where the dot represents the derivative with respect to conformal time, the prime represents

the derivative with respect to Q, and hk is the kth fourier mode of the perturbed metric. The

source term in Eq. (3) has several important properties. First, any realistic cosmological

model includes clustering matter components (baryons and dark matter), so hk must be

non-zero. Also, Q is non-zero. Hence, the source term on the right-hand side must be non-

zero overall. This is significant because it ensures that Q cannot be smoothly spread. Even

if Q is zero initially, the source term ensures that perturbations to grow.

A further consequence of the source term is that the perturbations in Q observed today

are extremely insensitive to the initial conditions for Q (Caldwell et al. 1998). Assuming

that Q/Q is comparable to the perturbations in other energy components, the transient

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solution to the perturbation equation is negligible today compared to the particular solution

set by the source term.

Why consider quintessence if its effect on the expansion of the universe is similar to

cosmological constant? The principle reasons are:

quintessence has different implications for fundamental physics;

quintessence may explain the cosmic coincidence problem;

quintessence may fit the observational data better than cosmological constant; and

quintessence may suggest a radically new picture of the overall history of the universe.

The first point is clear: whatever its identity, dark energy must be now incorporated in

any future attempt at a unified theory of fundamental interactions. A vacuum density or

cosmological constant () is static and spatially uniform. Its value is set once and for all

in the very early universe. Hence, is tied directly to quantum gravity physics near the

Planck scale. Quintessence is new dynamics at ultra-low energies (energy scale 1 meV

today), perhaps a harbinger of a whole spectrum of new low-energy phenomena. In addition,

for quintessence there is the added observational constraint that the coupling to ordinary

matter be sufficiently suppressed to evade fifth force and other constraints on light fields

(Carroll 1998).

The last three points are addressed in each of the remaining Sections.

II. FINE-TUNING, COSMIC COINCIDENCE, AND THE QUINTESSENTIAL

SOLUTION

Whatever form the dark energy takes, two new cosmological problems arise. First, the

component must have a tiny energy density today, roughly 10

47

GeV

4

. How does this smallvalue arise from a microphysical theory. We will refer to the puzzle of explaining this tiny

energy as the fine-tuning problem.

A second problem arises when the cosmological model is extrapolated back in time to the

very early universe, at the end of inflation, say. The quintessence energy density decreases at

a different rate than the matter density, and the ratio shrinks by many orders of magnitude

as we extrapolate back in time. The observations are telling us that, somehow, the ratio

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was set initially just right so that now, fifteen billion years later, the ratio is of order unity.

Accounting for the special ratio in the early universe will be referred to as the coincidence

problem (Steinhardt 1997). The coincidence problem is a generalization of the flatness

problem pointed out by Dicke and Peebles (Dicke & Peebles 1979).

The fine-tuning and cosmic coincidence problems are vexing. They are often posed as a

paradox: Why should the acceleration begin just as humans evolve? In desperation, some

cosmologists and physicists have been led to give renewed attention to anthropic models

(Weinberg 2000). But many continue to seek a dynamical explanation which does not

require the fine-tuning of initial conditions or mass parameters and which is decidedly non-

anthropic. A dynamical approach would seem to demand some sort of quintessence solution

since it would have to entail some interaction between the dark energy the matter-radiation

background.

At first blush, it would appear that replacing a cosmological constant with a scalar field

and potential energy is a step backwards. First, a general potential wont do. There must

be a value of such that V() equals todays dark energy density (1047 GeV4). Second,

we must explain why the field has that particular value today. In general, this is not simply

a matter of choosing the potential, but also a matter of carefully choosing the initial value

of the field and its time-derivatives. So, instead of tuning one parameter, the cosmological

constant, we must tune the parameters of the potential andthe initial conditions in the field.However, some creative solutions have been introduced to address the problem. We will

focus here on a single example which has combines several of these creative concepts. The

example, known as k-essence (Armendariz-Picon et al. 2000, 2001), is a form of quintessence

model in which the action for the scalar field has purely kinetic terms and no potential terms.

In addition to the canonical kinetic energy density term, X = 12()2, the k-essence action

has higher order non-linear kinetic couplings. The Lagrangian density can be written

L = 16R + 12 pk(X) + Lm (4)

where R is the Ricci scalar, and Lm is the Lagrangian density for dust and radiation and we

use units where 8G/3 = 1. The energy density of the k-field is k = (2Xpk,Xpk)/2; the

pressure is pk = pk/2; and the speed of sound of k-essence is c2s = pk,X/k,X . In string and

supergravity theories, non-standard kinetic terms appear generically in the effective action

describing the massless scalar degrees of freedom. Normally, the non-linear terms are ignored

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because they are presumed to be small and irrelevant. This is a reasonable expectation since

the Hubble expansion damps the kinetic energy density over time. However, one case in

which the non-linear terms cannot be ignored is if there is an attractor solution which forces

the non-linear terms to remain non-negligible. This is precisely what occurs here. Hence,

we wish to emphasize that k-essence models are constructed from building blocks that are

common to most quantum field theories. It is the dynamical attractor behavior (that often

arises in models with non-linear kinetic energy) which is responsible for the highly novel

features. The story is summarized in Fig. 1.

First, the nature of an attractor equation is that the evolution of the scalar field is

completely insensitive to the initial value of the field and its time derivatives. As indicated

in Fig. 1, the evolution of the dark energy component rapidly approaches an attractor

solution which depends only on the action itself. What is more remarkable is that the

attractor solution depends on what is happening in the rest of the universe. If the universe

is radiation-dominated, the k-essence behaves as if it were another radiation component

with w = 1/3, and its energy density decreases in parallel with the dominant radiation

component. Quintessence models with this property are called trackers, examples of which

include models with non-zero potentials (Zlatev et al. 1998, Steinhardt et al. 1999). So, not

only is the evolution independent of the initial conditions, but the tracking behavior insures

that the energy density of the k-essence field remains negligible compared to the radiationdensity throughout the radiation-dominated epoch. We have a dynamical explanation for

why the dark energy did not overtake the universe for the first 10,000 years. But, then,

something truly remarkable happens to k-essence models when the universe becomes matter-

dominated. The radiation-like attractor solution becomes unstable, and the energy density in

the k-essence field begins to drop several orders of magnitude until a new matter-dominated

attractor solution is found. This attractor solution keeps the k-essence density constant (see

Fig. 1), as ifw = 1. The drop in energy density means that the k-essence cannot dominate

immediately. But, once it hits the w = 1 attractor, the universe can only expand for

a short term before k-essence overtakes the universe and throws it into a phase of cosmic

acceleration.

In this scenario, the coincidence problem is beautifully addressed. Why did the universe

begin to accelerate just as humans evolve? Cosmic acceleration and human evolution are

both linked to the onset of matter-domination. The k-essence component has the property

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! # % &

' ) ' 0

" ' ' 3

4 ! # 57 6 6 6

6 6 6 ! 9 ' 9

' A ! & 0 " ' '

0D ! ' E )1 ' F A 3 H 6 6 6

6 6 6 A ' #

! 9 " ' )

P

P Q R T

P Q U

TP Q V

T

P Q W

T

P Q XT

P Q U V

T

` a b d

a f

d

g h

i p r

r s

t

u v x

y

y

y

y

FIG. 1: A plot showing the matter, radiation, and quintessence energy density as a function of

red shift for the case of k-essence models. The k-essence models are special cases of tracker

models with dynamical attractor solutions that funnel a wide range of initial conditions into a

common evolutionary track (upper left). The distinctive property ofk-essence is that the behavior

shifts at the onset of matter domination to an attractor solution that acts like a cosmological

constant (bottom). The bars below the graph indicate important events in cosmic history. The

late conversion to a cosmological constant behavior explains why cosmic acceleration has begun

only recently, a modest period (in terms of temperature scale) after matter domination.

that it only behaves as a negative pressure component after matter-radiation equality, so

that it can only overtake the matter density and induce cosmic acceleration after the matter

has dominated the universe for some period, at about the present epoch. And, of course,

human evolution is linked to matter-domination because the formation of planets, stars,

galaxies and large-scale structure only occurs after the matter-dominated epoch begins.At this point, the understanding of k-essence models is rather primitive, and the worked

examples are not very appealing in detail (too many terms and parameters). A better

understanding of non-linear attractor behavior is needed to see if simple, plausible examples

can be found. However, conceptually, k-essence is an important example of a dynamical,

non-anthropic explanation of the fine-tuning and cosmic coincidence problems that might

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arise from a fundamental theory.

III. DISTINGUISHING QUINTESSENCE FROM THE COSMOLOGICAL CON-

STANT

Distinguishing quintessence from a cosmological constant is a difficult challenge. We must

rely on the subtle differences between the two possibilities.

First, quintessence predicts a different value ofw and, hence, a different acceleration rate

than vacuum energy (w = 1). The effect is to change slightly the relation between angular

or luminosity distance and red shift. In Fig. 2, we compare two models with identical cosmic

parameters except that the value of w differs. Note that the position of the first acoustic

peak changes systematically as w changes. Of course, even more apparent are the changes inthe heights of the peaks. Unfortunately, neither effect can be used as a clear diagnostic for

distinguishing quintessence from a cosmological constant or determining w. As pointed out

by Huey et al., there is a degeneracy problem (Huey et al. 1999): a combination of variations

in m (the ratio of the matter density to the critical density), the Hubble parameter, the

curvature and w keep the CMB power spectrum nearly unchanged for constant w < 1/2.

Fig. 3 shows a dramatic example. As a result, if w is greater than 1/2 or rapidly time-

varying, then the microwave background only constrains a combination of parameters and

one must use other tests to resolve w independently.

A way of constraining w and the acceleration directly is by measurements of Type IA

supernova at deep red shift. Fig. 4 illustrates how well supernova measurements over a range

of red shift 0 < z < 2 can do in discriminating models with different constant w. The small

symbols represent what can be obtained by measuring thousands of supernovae with optimal

accuracy and using the systematic errors projected by the Supernova Acceleration Probe

(SNAP) team. One obtains an optimistic impression of how well w can be resolved. However,

caution is due. The points with the large error bars (towards the left) represent the first 40

supernovae that have been measured and their error bars. These are less impressive, offering

virtually no discrimination in the most likely range 1 w 2/3. The projections rely on

the assumption that the systematic errors are very small and that, by measuring thousands

of supernovae, the statistical errors can be reduced to the size of the small symbols.

Even with this assumption, there remains an additional degeneracy problem that cannot

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w = - 2/3

w = - 1

w = - 1/3

w = 0

0

2

4

6

8

10 100 1000

l(l+1)C

l/2

(x1

0-10

K2)

multipole moment (l)

FIG. 2: The cosmic microwave background power spectrum (multipole moments C vs. multipole

number ) for a sequence of models with identical parameters except for varying w. Note the small

shift in the position of the first acoustic peak, as indicated by the tilted line.

0 200 400 600 800 1000multipole moment (l)

0

5

10

l(l+1)C

l/2

(x10-10

K2)

FIG. 3: An illustration of the cosmic degeneracy problem: the cosmic microwave background power

spectrum for a flat quintessence model with w = 0.56, m = 0.30, Q = 0.7, and h = 0.56 (thick

dashed0 and for a closed model with w = 1, m = 0.35, = 0.7 and h = 0.6 (thin solids). At

the resolution of this figure, the two curves completely overlap

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0.0 0.5 1.0 1.5 2.00.2

0.0

0.2

0.4

0.6

0.8

relativemagnitu

de

red shift

matter only

(no quintessence)

w = -1/3

w = -2/3

w = --1

FIG. 4: Magnitude versus red shift relative to a model with no quintessence (m = 1) for m = 0.3

and w = 1, 2/3, 1/3 and 0. Small symbols represent the optimal that can be achieved by

measuring thousands of supernovae and binning by red shift. Data point with error bars (on left)

represent current data from individual galaxies.

be resolved. Namely, if we do not assume w(z) is constant, but, instead, consider the pos-sibility that w varies with redshift (as in k-essence models, for example), then the ability of

supernovae surveys to resolve w today or its time-variation is enormously reduced. Fig. 5

shows a group of models with widely varying w and dw/dz today, along with the corre-

sponding predictions for luminosity distance dL(z). The figure illustrates a fundamental

degeneracy that makes it difficult to resolve w to much better than 40% or to obtain any

useful information about dw/dz (Maor et al. 2001). The interested reader should consult

Maor et al. 2002 to see more illustrations and details of this degeneracy problem.

At present, I am unaware of any probe of combination of probes that can precisely

determine w and its time-variation (Maor et al2002). Predictions of high resolution are all

based on the assumption that dw/dz is constant. Many models satisfy this approximation,

and so the tests can be useful for selecting out some possibilities. But, a key challenge in

the field is to find a better, more general test.

Another distinctive property of quintessence is that it is spatially inhomogeneous. Eq. (3)

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1.0

2.0

3.0

H0d

L(z)

0.0 0.5 1.0 1.5 2.0

red shift (z)

-1.2

-0.7

-0.2

wQ

(z)

(a)

(b)

FIG. 5: An illustration of the supernova degeneracy problem: The luminosity distance dL(z) vs.

red shift curves are nearly degenerate in (a) for the nine very different choices of quintessence

equation-of-state wQ(z) shown in (c). All models have m = 0.3 and H0 is the current value of

the Hubble parameter.

can be used to predict the fluctuations in the quintessence energy density. The biggest effect

is on the large angular scale microwave background anisotropy, because the quintessence fluc-

tuations are weak compared to the matter fluctuations at smaller scales and the quintessence

energy density is negligible when those length scale enter the horizon.

On large angular scales, quintessence fluctuations can alter the the low multipole moments

of the cosmic microwave background power spectrum. This modification is in addition to

the usual (late) integrated Sachs-Wolfe (ISW) effect. The ISW occurs in any model with

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1020.5

1.0

1.5

2.0

l(l+1)C

l/2

(x

10-10

K2)


FIG. 6: A comparison of the large angular scale microwave background anisotropy for a model

with cosmological constant (dashed), and quintessence models with w = 2/3 (solid), w = 1/3

(dot-dashed) and rapidly time-varying w with w = 0 at present (dotted).

m < 1, whether an open model, a model with quintessence or a model with a cosmological

constant. It comes about because the gravitational potential well due to a mass fluctuation

changes as a CMB photon traverses the fluctuation passing from the last scattering surface

to the present. The net ISW effect is an increase in the multipoles on angular scales which

enter the horizon when m < 1, that is to say, the low- multipole moments. Fluctuations in

the quintessence component cancel this effect because they add to the gravitational potential

(Dave et al. 2002).

Even at the largest angular scales, the fluctuation effect is weak, becoming completely

negligible as w approaches -1. Fig. 6 shows the low multipole moments (large angular scale

anisotropy) of the microwave background temperature power spectrum for models withconstant and time-varying w. Unfortunately, the fluctuation effect is very small unless w

1/3 or very rapidly time-varying, which is inconsistent with other cosmological constraints.

Another way to distinguish the nature of dark energy is to measure its sound speed to

determine if it is different from unity. The sound speed can be detected because it also

affects the perturbations in the quintessence energy distribution. This approach is less

generic because the sound speed in many models of quintessence in the literature is equal

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to unity (the speed of light), e.g., models in which quintessence consists of a scalar field ()

with canonical kinetic energy density (X 12

()2) and a positive potential energy density

(V()). However, in general, the sound speed can differ from unity and vary with time, as is

the case for k-essence models (see also Carturan & Finelli, 2002). Detecting these effects is

an independent way of showing that dark energy does not consist of a cosmological constant.

Fig. 7 illustrates models in which the sound speed varies. See DeDeo et a. 2003 for

details. When the sound speed is near zero, there can be significant effect even w is close to

-1, consistent with current constraints on w. The effects on the acoustics peaks and higher

multipole moments are also relevant. If the quintessence density is at least one percent of

the critical density at the surface of last scattering (as is the case for many tracker and

k-essence models, for example), the modifications of the heights and shapes of the acoustic

peaks in models where the sound speed is near zero are small but distinguishable from the

effects due to variations of other cosmic parameters, as shown by Erickson et al. (2002).

In addition, the sound speed can produce oscillations and other Effects in the mass power

spectrum, as shown in Fig. 9.

The highly precise data obtained from the MAP and Planck satellites and from the Sloan

Digital Sky Survey may reveal these subtle effects. The precise behavior of w and the sound

speed is, by itself, of limited interest. But, what is extraordinarily important about the

detection of any deviation from c2s = 1 is that it would be a direct sign that the dark energy

is a complex, dynamical fluid rather than an inert cosmological constant. Hence, it is a

target well worth pursuing.

IV. DARK ENERGY PAST AND FUTURE

Most cosmologists have treated dark energy as a simple modification of the standard

big bang/inflationary picture. But, maybe its discovery signals the need to re-evaluate our

overall understanding of cosmic evolution.

Today, the consensus model of our cosmic history is based on the big bang picture com-

bined with inflationary cosmology. This model has been subjected to an extraordinary

battery of cosmological tests in the past decade, ranging from measurements of the cosmic

microwave background to detailed surveys of large scale structure. The original picture,

based on the Einstein-de Sitter model (a flat universe with matter density equal to the

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0 10 20 30 40 500

1

2

l(l+1)C

l/2

(x10-10

K2)


FIG. 7: Comparison of the lowest multipole moments of the CMB temperature power spectrum

for a series of models with the same w(z). but different cs(z): (a) cs = 1 (dotted); (b) cs = 1 for

z > 10 and cs = 0 for z < 10 (solid); and a sequence of k-essence models (dot-dashed, short- and

long-dashed).

critical density), failed many of the tests, but replacement of 70% of the dark matter witha gravitationally self-repulsive dark energy, produced a new consensus model in exquisite

agreement with all cosmological tests. Hence, many cosmologists are prepared to declare

our cosmic history a settled issue.

However, a second look suggests some cause for concern. The new consensus model now

requires two periods of accelerated expansion: one in the early universe, corresponding to a

rate in which the universe doubles in size every 1035 seconds, and now a second, in which

the doubling rate is fifty orders of magnitude less. Each period of acceleration requires its

own energy source which must be finely-tuned to satisfy observational constraints. The first

acceleration has a well-defined purpose, to homogenize and flatten the universe. The second

was not predicted by either the big bang or inflationary pictures and it plays no known role

in the universe. (Of course, now that we know the matter density is less than the critical

density, we need dark energy to bring the total to the critical value, predicted by inflation.

However, the expectation had originally been that the matter density would itself equal the

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00

2

4

6

8

500 15001000

l(l+1)C

l/2

(x10-10

K2)


FIG. 8: Comparison of higher multipole moments of the CMB temperature power spectrum for

the models in Fig. 7. The spectra have been normali d so that the amplitudes match at the top of

the first acoustic peak.

critical density and that there is no dark energy.)

The recent proposal of a cyclic universe presents a whole new outlook on cosmic history

in which dark energy plays a central role (Steinhardt & Turok, 2002a, 2002b). in this model,

the conventional cosmic history is turned topsy-turvy. The big bang is not the beginning

of time. Rather, it is a bridge to a pre-existing contracting era. The Universe undergoes

a sequence of cycles in which it contracts in a big crunch and re-emerges in an expanding

big bang, with trillions of years of evolution in between. The big bang is moderated. The

temperature and density of the universe do not become infinite at any point in the cycle;

indeed, they never exceed a finite bound (about a trillion trillion degrees). No inflation has

taken place since the (last) bang. The current homogeneity and flatness were created by

events that occurred before the most recent big bang,and the seeds for galaxy formation

were created by instabilities arising as the Universe was collapsing towards a big crunch,

prior to our big bang.

In this picture, dark energy is moved to center stage and is part of the engine that drives

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10-3

10-2

10-1

k (h-1

Mpc-1

)

10-1

100

101

P(k)(arb.normalization)

FIG. 9: Comparison of the shape of the total fluctuation power spectrum, P(k) as a function of

wavenumber k for the sequence of models in Fig. 7. The normalization of the curves is arbitrary.

the periodic evolution of the universe. Dark energy recurs as the dominant form of energy

every cycle roughly 15 billion years after each bang. and it replaces two of the key roles of

inflation. Although it causes the universe to accelrate at an pace 100 orders of magnitudeslower than inflation, by maintaining the acceleration for a trillion years or so, the dark

homogenizes and flattens the universe. In particular, it is the dark energy of a cycle ago

that made the universe homogeneous and flat prior to our own big bang,

A second critical feature of the dark energy is that it is not stable. It naturally decreases

with time as the universe expands. As a result, the acceleration ultimate stops and the

universe begins to decelerate. It eventually triggers a period of contraction, during which

there is the quantum generation of a nearly scale-invariant spectrum of perturbations that

accounts for the temperature fluctuations of the cosmic microwave background and large

scale structure.

Finally, the dark energy is responsible for insuring that the cyclic evolution is an attractor

solution to the evolution equations. If random fluctuations kick the universe away from the

ideal cyclic evolution, the period of dark energy domination red shifts away the transient

behavior and drives the universe back towards the regular cyclic solution.

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To complete the picture, we should note that the cyclic model is motivated by recent

developments in string theory, especially the ideas of branes and extra dimensions. In

this picture, our three-dimensional universe may be a hypersurface embedded in a space

with one or more extra-dimensions. In a version of string theory known as M-theory, for

example, this hypersurface (a membrane-like surface known as a brane) constitutes one of

the boundaries of the extra dimension, and another brane lies at the other boundary. The

cyclic model proposes that the two branes interact with one another through gravity and the

exchange of virtual strings and branes, resulting in a weak force that causes the branes to be

drawn together and collide at regular integrals. Each collision causes the branes to bounce

back to their original positions and creates matter and radiation whose gravitation causes

the branes to begin to stretch. This represents the bang and the subsequent expansion and

cooling. The expansion continues at a decelerating rate until, after 15 billion years, the

matter and radiation density are so thinly spread that it becomes negligible compared to

the potential energy of the interbrane force. This potential is the dark enegy that drives

the period of acceleated expansion that has recently been observed. The branes stretch

at an accelerating rate, thinning the matter density to a near vacuum and flattening any

curvature and warps in the branes. Eventually, the weak force draws the brane together,

reducing the dark energy and naturally ending the accelerated expansion. The contraction

that ensues is the contraction of the extra dimension. Our three-dimensions (the branes)remain stretched out and the temperature and density remain nearly zero until the branes

collide. And the cycle continues.

V. FINAL REMARK

The discovery of the retrograde motion of Mars was a surprise that was originally ex-

plained as a minor modification of the heliocentric model. Only after Copernicus, Kepler

and Newton was it recognized as the first hint of a great scientific revolution.

Perhaps there is a lesson here. The observation of cosmic acceleration has forced us to

revise the big bang/inflationary picture. Should we believe, as most cosmologists suggest,

that this is the last missing piece of the puzzle and our understanding of the universe is

virtually complete? Or have we just uncovered a deep dark secret that that will revolutionize

our whole view of the universe and our place in it? I must confess to my own prejudice that

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the latter seems more likely.

I would like to thank the Organizing Committee and the Royal Society for offering me

the opportunity to participate in this symposium. I also want to express my appreciation to

the many students and colleagues who have guided my thinking and with whom I have had

the pleasure of collaborating in the works cited here, including R. Brustein, R. Caldwell,

R. Dave, S. DeDeo, J. Erickson, G. Huey, J. Khoury, I. Maor, J. McMahon, J. Ostriker, B.

Ovrut, N. Turok, and L. Wang. This work was supported in part US Department of Energy

grant DE-FG02-91ER40671 (PJS).

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paul j. steinhardt- a quintessential introduction to dark energy

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